An optimal control theory approach to non-pharmaceutical interventions

Lin, Feng; Muthuraman, Kumar; Lawley, Mark

If the cost function is linear, the control policy is bangbang, which suggests implementing NPIs at the maximum strength or not implementing at all as shown in Figure 2 . If the cost function is nonlinear, for example the quadratic cases presented in Figure 4 , the control policy has multiple levels, which requires varying the NPI strengths as the system evolves from one state to another. It is easily shown from Eqs. 7 and 9 that if the linear model indicates NPI implementation for a state x, i.e., u* = b for state x, then the quadratic model also indicates some implementation in state x, i.e., u* > 0 for state x. But the inverse is not true. This property can be proved easily. If u* = b in the linear model, we have ψ(s, i) < 0, for all (s, i) Ω1 according to Eq. 7. So